between them defined as follows: for \(1\le \ell \le p+1\)  and \(\sigma =\lbrace  j_1 < \cdots < j_{p+1} \rbrace \)  consider \(\partial _\ell \sigma \lbrace  j_1 , \ldots , \hat{\jmath }_\ell , \ldots , j_{p+1} \rbrace \) , obtained by removing \(j_\ell \) ; then, for \(p\ge 1\) , observe that the inclusions \(C_*(\cap _{j\in \sigma }X^{j})\hookrightarrow C_*(\cap _{j\in \partial _\ell \sigma }X^{j})\)  induce chain maps \(\delta ^{(\ell )}_{p,*} :C_{p,*} \rightarrow C_{p-1,*}\)  and define \(\delta _{p,*}  \sum _{i=0}^p (-1)^{i} \delta _{p,*}^{(p+1-i)}\) .
A chain map \(\varepsilon _* = \delta _{0,*}:C_{0,*} \rightarrow C_*(\cup _{j\in J} X^j)\)  is induced in a similar way by the inclusions \(C_*(X^{j})\hookrightarrow C_*(\cup _{j\in J} X^{j})\) .
Even if in the definition of \(\delta _{p,*}\)  we used a different choice from [1]} for the alternating signs, this leads to an isomorphic construction of the double complex.
